Choose 10,000 random points inside mesh
There is  way to do this without python, using particles.


Create a particle system and pick volume and random as the distribution settings:



Also set the physics to none to conserve processing power.

You'll want to set the Number to 10000, but I don't think the computer I'm currently on will like that :P.
Create a one-vertex object. The easiest way to do this is by adding a plane and deleting three vertices. After you delete the extra verts, switch back to object mode (Tab) and press ShiftCtrlAltC> Origin to geometry to snap the origin to the single vertex.
Pick your single vertex object as the particle system object in Particles > Render:


With the object with the particle system selected, press Convert on the particle modifier in the  modifiers tab:


Join the resulting vertices to your object with CtrlJ (everything should already be selected correctly). Remember to remove the particle system, as pressing Convert does not delete it.


Result: